Show your work please.

You and your roommate are passionate about designing customized T-shirts, and you plan on opening an online store to sell them. Your roommate knows you took BUAD 311 last semester so he assigns you to oversee pricing. Based on sales data from similar online T-shirt stores, you estimate demand to be a function of the price p:

D(p) = 600 - 15p, 0<=p<=40.

It costs $2 to produce each T-shirt.

a) For a single-price policy, what is the optimal price to set in order to maximize the total profit? What is the optimal profit?

Answer: The optimal price is $21 and optimal profit is $5,415.

b) Your friend tells you that at most 180 T-shirts can be made. With this new information in mind, how should you adjust the optimal price from a)? What is the new profit?

Answer: The new optimal price should be $28.

The new profit is $4,680.

c) (Ignoring part b). Realizing that price differentiation might bring more revenue, you plan to divide the market into 2 different segments. For the customers with high willingness to pay, you offer a price of $10; for the customers with low willingness to pay, you offer a price of $5. What is the new profit?

Answer: Total profit 450*8 + 75*3 = 3825

d) Following from part c), suppose we fix the price for the customers with high willingness to pay at $10. For the customers with willingness to pay lower than $10, what is the optimal price to set for them? How much does the total profit change compared to c)?

Answers: Optimal low price = $6

Optimal profit 450*8 + 60*4 = 3840

Increases by $15